Verifying Two Conjectures on Generalized Elite Primes

Transcription

1 Journal of Integer Sequences, Vol. 12 (2009), Article Verifying Two Conjectures on Generalized Elite Primes Xiaoqin Li 1 Mathematics Deartment Anhui Normal University Wuhu , Anhui Peole s Reublic of China qinlxss@163.com lxq623@mail.ahnu.edu.cn Abstract A rime number is called b-elite if only finitely many generalized Fermat numbers F b,n = b 2n + 1 are quadratic residues modulo. Let be a rime. Write 1 = 2 r h with r 0 and h odd. Define the length of the b-fermat eriod of to be the minimal natural number L such that F b,r+l F b,r (mod ). Recently Müller and Reinhart derived three conjectures on b-elite rimes, two of them being the following. (1) For every natural number b > 1 there is a b-elite rime. (2) There are generalized elite rimes with elite eriods of arbitrarily large lengths. We extend Müller and Reinhart s observations and comutational results to further suort above two conjectures. We show that Conjecture 1 is true for b and that for every ossible length 1 L 40 there actually exists a generalized elite rime with elite eriod length L. 1 Introduction The numbers of the form F b,n = b 2n + 1 are called generalized Fermat numbers (GFNs) for natural numbers b and n. The definition generalizes the usual Fermat numbers F n = 2 2n + 1, which were named after Pierre Simon 1 Research suorted by NSF of China Grant

2 de Fermat ( ). A lot of research has been done on Fermat numbers and their generalization since then (see [2, 6, 7, 8]). In 1986 Aigner [1] called a rime number elite if only finitely many Fermat numbers F n are quadratic residues modulo, i.e., there is an integer index m for which all F n with n > m are quadratic non-residues modulo. He discovered only 14 such rime numbers less than More comutational effort yielded all 27 elites u to together with some 60 much larger numbers [3, 4, 9]. These rime numbers are summarized in sequence A of Sloane s On-Line Encycloedia of Integer Sequences [13]. Müller and Reinhart [10] generalized Aigner s concet of elite rimes in analogy to that of Fermat numbers. Definition 1.1. ([10, Definition 1.1]). Let be a rime number and b 2 be a natural number. Then is called a b-elite rime if there exists a natural number m, such that for all n m the GFNs F b,n are quadratic non-residues modulo. By the recurrence relation F b,n+1 = (F b,n 1) 2 + 1, (1) one sees that the congruences F b,n (mod ) eventually become eriodic. Write 1 = 2 r h with r 0 and h odd. Then this eriod Müller and Reinhart [10] called it b-fermat eriod of begins at latest with the term F b,r. So there has to be a minimal natural number L such that F b,r+l F b,r (mod ), (2) which they [10] call the length of the b-fermat eriod of. The terms F b,n (mod) for n = r,...,r + L 1 are the b-fermat remainders of. Therefore, a rime number is b-elite if and only if all L b-fermat remainders are quadratic non-residues modulo. It is moreover known that for all it is a necessary condition for eliteness with L > 1 that L is an even number smaller than +1 (comare [10]). 4 Müller and Reinhart [10] gave fundamental observations on b-elite rimes and resented selected comutational results from which three conjectures are derived, two of them being the following. Conjecture 1. [10, Conjecture 4.1] For every natural number b > 1 there is a b-elite rime. Conjecture 2. [10, Conjecture 4.2] There are generalized elite rimes with elite eriods of arbitrarily large lengths. Concerning Conjecture 1, Müller and Reinhart [10] observed that most of the bases b actually have the rime 3 or 5 as b-elite only the bases b 0 (mod 15) do not belong to one of these two trivial families. Conjecture 2 seems to be suorted by their comutations. They [10] roved the following Lemma 1.1. Lemma 1.1. For every L L 1 = {1, 2, 4, 6, 8, 10, 12}, (3) there is a generalized elite rime < 10 4 with elite eriod length L. 2

3 The main urose of this aer is to extend Müller and Reinhart s observations and comutational results to give further suort to the two conjectures above. We state our main results as the following two Theorems. Theorem 1. Conjecture 1 is true for 1 < b More recisely, for every natural number 1 < b 10 13, there is a b-elite rime Theorem 2. For every L {1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40}, there is a generalized elite rime with elite eriod length L. In Section 2 we give an algorithm to test the b-eliteness of for given b 2 and rime. The main tool of our algorithm is the Legendre symbol. Comarison of effectiveness with Müller s method for testing the 2-eliteness of is given, see Remark 2.2. In Section 3 we rove Theorem 1. We first roose a sufficient and necessary condition on base b 2 to which there is a b-elite rime {3, 5, 7, 11, 13, 19, 41, 641}. Using the condition and the Chinese Remainder Theorem, it is easy to comute a set R with cardinality R = such that Conjecture 1 is already true for those bases b such that b ( mod m) R, where m = = Thus we only need to consider the bases b with b (mod m) R. We then give an algorithm to find the smallest b-elite rime P b for each base b = um + b i with b i R. At last we tabulate P(B) and the smallest b with P b = P(B) for B = 10 10, 10 11, 10 12, 10 13, where P(B) is defined by (15) in section 3. In articular, we have P(10 13 ) = = P Theorem 1 follows. In Section 4 we rove Theorem 2. At first we comute all the elite eriods of every generalized elite rime < 10 7 based on the method described by Müller and Reinhart [10]. As a result we find some elite eriod lengths L L 2 = {14, 16, 18, 20, 22, 24, 26, 28, 30, 36}. (4) For every L L 2, we tabulate P(L) and the smallest b to which P(L) is elite with length L, where P(L) is defined by (17) in section 4. In articular we have P(36) = (the smallest base b = 5369). We also give a new method to find some elite rimes with elite eriod lengths 32,34,38 and 40, where L = 40 is realized by the elite rime = (the smallest base b = 54712). Thus Theorem 2 follows. 2 A b-eliteness testing algorithm Let b > 1 be an integer, and let = 2 r h + 1 be a rime number with r 1 and h odd. In this section, we will give an algorithm to test the b-eliteness of. Let ( ) denote the Legendre symbol. Our algorithm is based on the following criterion. Fb,n F b,n is a quadratic non-residue modulo if and only if = 1. (5) 3

4 Given b 2 and rime = 2 r h + 1 with h odd, we check whether Fb,n = 1 (6) holds for n = r,r+1,r+2,... consecutively, where F b,n ( mod ) are comuted recursively by (1). If (6) does not hold for some n r, then is not b-elite. If (6) holds for r n r+l 1, then is b-elite, where L is the length of the b-fermat eriod of, namely the least ositive integer such that (2) holds. Now we describe our Algorithm 2.1 in the following seudocode. Algorithm 2.1. Testing the b-eliteness of rime ; {Inut b 2 and rime } {Determine whether is b-elite or not; if is b-elite then outut the length L} Begin Finding r and h such that = 2 r h + 1 with h odd; f b F b,r (mod ); f f b ; L 0; elite True; f Reeat Comuting b by [5, Algorithm 2.3.5] (cf. also [12, 11.3]); If 1 Then elite False Else f b begin f b (f b 1) (mod ); L L + 1 end; Until (not elite) or (f b = f); If elite Then outut L Else outut is not b-elite End. Remark 2.1. The rime 2 is not b-elite to any b 2 since there is no quadratic non-residue modulo 2. So here and for the rest of this aer, we only need to consider odd rimes. Remark 2.2. Let q be a rime and c be a ositive integer with q c. Denote by ord q (c) the multilicative order of c ( mod ). Müller [9] gave an eliteness testing algorithm [9, Algorithm 3.1] for the base b = 2 based on the following criterion [9, Theorem 2.1]. F 2,n is a quadratic non-residue modulo if and only if 2 r ord (F 2,n ). (7) To check whether 2 r divides ord (F 2,n ) for n = r,r + 1,r + 2,..., the algorithm comutes F 2k h 2,n (mod) for k = 0, 1,...,k 0, where k 0 = min{0 k r : F 2k h 2,n 1 (mod)}. It is well-known [5, 2.1.2] (see also [12, Theorem 4.9]) that it takes O(lns ln 2 ) bit oerations to comute the modular exonentiation F2,n s (mod ). With our method, we ) comute the Legendre symbol, which requires only ( F2,n O(ln 2 ) bit oerations [5, 2.3] (see also [12, Corollary and Exerice ]). So, for testing the b-eliteness of, using criterion (5) is faster than using criterion (7). 4

5 3 Proof of Theorem 1 Throughout this section, let N = {0, 1, 2, 3,...} be the set of all natural numbers. Let B A be two sets. We denote by A the number of elements in A, and Let be an odd rime. Define the sets and A B = {c : c A,c B}. A = {0, 1, 2,..., 1}; { {b( 2) A : is b-elite} {1}, if is ( 1)-elite; B = {b( 2) A : is b-elite}, if is not ( 1)-elite; R = A B. To rove Theorem 1, we need nine Lemmata. Lemma 3.1. [10, Observation 2.2,2.3] Let be an odd rime number, b be a natural number. If is b-elite, then (1) is (b + k)-elite for k {a,a + 1,a + 2,...}, where a = b ; (2) is ( b)-elite if 2 b <. Moreover, the Fermat remainders and the resective length of the Fermat eriod for the bases b + k and b are the same. By Lemma 3.1 we have Lemma 3.2. Let be an odd rime and b (> 1) N. Then is b-elite if and only if b (mod ) B. Lemma 3.3. [10, Consequence 2.10] We have R 3 = R 5 = {0}. Lemma 3.4. [10, Theorem 2.13] Let b be a natural number and be an odd rime number. ( Then ) is b-elite with L = 2 if and only if ( ) 7 (mod 12) and either b 2 +1 b (mod ) with = 1 or b b (mod ) with = 1. b Lemma 3.5. We have R 7 = {0, 1, 6}. Proof. Let k {0, 1, 2, 3, 4, 5, 6}. Then and b k, if k = 3, 5; k 2 7 k, if k = 2, 4; + 1 (mod 7) = 1, if k = 0; 2, if k = 1, 6; k = 7 1, if k = 1, 2, 4; 0, if k = 0; 1, if k = 3, 5, 6; Based on Lemma 3.4, we have B 7 = {2, 3, 4, 5}. Thus the Lemma follows. 5

6 Using Algorithm 2.1, we can easily get the following Lemma 3.6 and Lemma 3.7. Lemma 3.6. We have R 11 = {0, 2, 3, 4, 5, 6, 7, 8, 9}; R 13 = {0, 2, 3, 4, 6, 7, 9, 10, 11}; R 19 = {0, 2, 3, 4, 5, 6, 9, 10, 13, 14, 15, 16, 17}; R 17 = A 17 and R 23 = A 23. Lemma 3.7. Let < 1000 be an odd rime. Then where R < 1 2 A if and only if {3, 5, 7, 41, 641}, R 41 = {0, 1, 3, 9, 14, 27, 32, 38, 40} and R 641 = {0, 1, 2, 4, 5, 8, 10, 16, 20, 21, 25, 29, 31, 32, 40, 42, 50, 58, 61, 62, 64, 67, 77, 80, 84, 100, 105, 116, 122, 124, 125, 128, 129, 134, 141, 145, 153, 154, 155, 159, 160, 168, 177, 199, 200, 210, 221, 232, 241, 243, 244, 248, 250, 256, 258, 268, 282, 287, 290, 305, 306, 308, 310, 318, 320, 321, 323, 331, 333, 335, 336, 351, 354, 359, 373, 383, 385, 391, 393, 397, 398, 400, 409, 420, 431, 441, 442, 464, 473, 481, 482, 486, 487, 488, 496, 500, 507, 512, 513, 516, 517, 519, 525, 536, 541, 557, 561, 564, 574, 577, 579, 580, 583, 591, 599, 601, 609, 610, 612, 616, 620, 621, 625, 631, 633, 636, 637, 639, 640}. Remark 3.1. In fact, the two results R 17 = A 17 and R 23 = A 23 were already given by Müller and Reinhart [10] where they are immediate consequences of Theorem 2.15 and Theorem By Lemma 3.6 and Lemma 3.1, the rimes 17 and 23 are not b-elite to any natural number b 2. Let m = = (8) Alying the Chinese Remainder Theorem, it is easy to comute the set which has cardinality R = {0 b < m : b (mod) R for = 3, 5, 7, 11, 13, 19, 41, 641} (9) R = R = = (10) Using the Hea Sort Algorithm [11, 8.3], we sort the elements of R in an increasing order where R = {b 1 < b 2 < < b R }, (11) b 1 = 0,b 2 = 1590,b 3 = 2955,b 4 = 5685,b 5 = 6405,b 6 = 7020,...,b R = By Lemma 3.2 and the Chinese Remainder Theorem we have the following Lemma

7 Lemma 3.8. Let b (> 1) N, and let m be given as in (8). Then there is a b-elite rime {3, 5, 7, 11, 13, 19, 41, 641} if and only if b (mod ) B for some {3, 5, 7, 11, 13, 19, 41, 641} if and only if b (mod m) R. From Lemma 3.8, we see that Conjecture 1 is already true for those bases b such that b (mod m) R. Thus we only need to consider the bases b with b (mod m) R. Lemma 3.9. For every base 1 < b with b (modm) R, there is a rime such that is b-elite. Proof. Given b 2 with b (mod m) R, let P b = {rime : is b-elite}, (12) and let P b = {, if P b =, min{ : P b }, otherwise. Let 1 B 1 < B 2 be two natural numbers such that (13) R {b (mod m) : B 1 < b B 2 }. Define the functions and P(B 1,B 2 ) = max{p b : B 1 < b B 2,b(mod m) R}, (14) P(B 2 ) = P(1,B 2 ) = max{p(1,b 1 ),P(B 1,B 2 )}, (15) With the above rearation, we describe our algorithm for verifying Lemma 3.9 for those bases b with B 1 < b B 2 and b (mod m) R. Algorithm 3.1. Verifying Lemma 3.9 for B 1 < b B 2 with b (mod m) R; {Inut B 1,B 2 and max with 1 B 1 < B 2, say B 1 = 10 10, B 2 = 10 11, and max = 10 9 } {Outut either P = P(B 1,B 2 ) < max and the smallest b (B 1,B 2 ] such that P = P b } {or the smallest b (B 1,B 2 ] with b (mod m) R such that Lemma 3.9 fails} Begin P 3; u B 1 m; m b u; j 1; First True; While b B 1 Do begin j j + 1; b u + b j end; Reeat If First Then begin i j; First False end Else i 0 ; reeat 29; Found False; Reeat b b (mod ); Using Algorithm 2.1 to test the b -eliteness of ; If is b -elite Then begin Found True; If > P Then Begin P ; b b End end Else begin the next rime > ; If ( = 41) or ( = 641) then the next rime > 7

8 end Until Found Or ( > max); If not Found Then begin outut the lemma fails at b, enlarge max and try again ; exit end; i i + 1; b u + b i ; until (i > R) or (b > B 2 ); u u + m; Until u > B 2 ; Outut P and b; End; The Delhi rogram ran about 105 hours on a PC AMD 3000+/2.0G to find P(1, ), P(10 10, ), P(10 11, ), and P(i 10 12, (i + 1) ), for i = 1, 2,...,9. Then by (15) we get P(B) for B = 10 10, 10 11, 10 12, ; see Table 1, where b is the first base with P b = P(B). As a result we have P(10 13 ) = , which means that for every base 1 < b with b (modm) R, there is a rime such that is b-elite. The Lemma follows. Lemma 3.9 together with Lemma 3.8 imlies Theorem 1. Table 1: P(B) and b with P b = P(B) B P(B) b L Proof of Theorem 2 Let L {1, 2, 4, 6, 8,...}. Define P(L) = {rime : is a generalized elite with eriod length L} (16) and let P(L) = {, if P(L) =, min{ : P(L)}, otherwise. (17) 8

9 By Lemma 1.1, Müller and Reinhart [10] have found P(L) for L L 1 (L 1 is given by (3)). We summarize their comutations in the following Table 2, where b is the base to which P(L) is elite with elite eriod length L. Table 2: The function P(L) for L L 1 L P(L) b To rove Theorem 2, we need three Lemmata. Lemma 4.1. [10, Theorem 2.18] Let = 2 r h + 1 with h odd. Let n be the number of all ossible eriods and denote by L i the length of the eriod i. Then n L i = h. (18) i=1 The number N b,i of all b s in the eriod i is N b,i = 2 r L i. (19) Lemma 4.2. For every L L 2 (L 2 is given by (4)), there is a generalized elite rime < 10 7 with elite eriod length L. Proof. Based on Lemma 4.1, Müller and Reinhart [10] resented an algorithm to find the first elite eriod of rime. Given a rime = 2 r h + 1 with h odd, let Then we have S 2 r = h and S 2 r = {c Z : b Z such that b 2r c (mod )}. (20) F b,n 1 (mod ) S 2 r (21) for n = r,...,r + L 1 and for all bases b. Moreover, elements of S 2 r belong to many different eriods of various lengths. Let g be a rimitive root modulo and let c S 2 r. Then we have c = g 2r k 0 with k 0 {0, 1,...,h 1}. Let g 0 = g k 0. We check whether g0 2r+i + 1 = 1 (22) holds for i = 0, 1, 2,... consecutively. If (22) does not hold for some i, then this Fermat eriod is not an elite eriod. If (22) holds for 0 i < l, where l is the smallest natural 9

10 number such that g 2 r+l 0 g 2 r 0 (mod ), then the eriod c + 1,c 2 + 1,...,c 2l is an elite eriod with length L = l. It is easy to modify Müller and Reinhart s comutational method in order to find all elite eriods of every generalized elite rime < Bound, say Bound = Now we describe the modified algorithm in the following seudocode. Algorithm 4.1. Finding all elite eriods L of each rime < Bound if they exist. {Inut Bound, say Bound = 10 7 ; Outut < Bound with L > 12.} Begin 3; Reeat Finding r and h such that = 2 r h + 1 with h odd; g rimitive root mod ; For i 0 To h Do tested i False; eriodstart 0; reeat index eriodstart; elite T rue; L 0; Reeat tested index True; If elite Then begin f g 2r index + 1 (mod ); f Comuting by [5, Algorithm 2.3.5] (cf. also [12, 11.3]); If 1 Then elite False; f end; index index 2 (mod h); L L + 1 ; Until (index = eriodstart); If elite and (L > 12) Then outut and L; While tested eriodstart Do eriodstart eriodstart + 1; until (eriodstart = h); the next rime > ; Until > Bound End. The Dehi rogram ran about 53 hours to comute all elite eriods of every elite rime < 10 7, and find some elite eriod lengths L L 2. For every L L 2, we summarize P(L) and the smallest b to which P(L) is elite with length L in Table 3. The Lemma follows. Table 3: The function P(L) for L L 2 L P(L) b Remark 4.1. The smallest base b in Table 3 can be easily obtained by using Algorithm 2.1 to test the b-eliteness of P(L) for b = 2, 3,..., P(L) 1 consecutively until the length of the 2 elite eriod is found to be L. Remark 4.2. There are no generalized elite rimes < 10 7 with L = 32 or 34 or L >

11 In the following Table 4, we list the factorization of 2 L and the factorization of P(L) 1 for L L 3 = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 36} Table 4: The factorizations of 2 L and of P(L) 1 for L L 3 L P(L) The factorization of 2 L The factorization of P(L) Let L be even. Define Then for every L L 3, we find that, q L = max{rime q : q 2 L 2 + 1}. q L (P(L) 1). (23) Based on (23), we try to find some generalized elite rimes with elite eriod lengths 32,34,38 and 40. The method is as follows (taking L = 32 for examle). Since 2 L = = = = F 4, we have q 32 = In order to find the elite rime with length 32, we consider rimes ( > 10 7 ) which can be written in the form = 65537k + 1 with k an integer. Using Algorithm 4.1, we comute all the elite eriods of these elite rimes consecutively until the length of the elite eriod is L. As a result we find that rime is b-elite with L = 32, where b = and = In Table 5, for L = 32, 34, 38 and 40, we tabulate the rime, the base b to which is elite with length L. 11

12 Table 5: Elite rimes with length L = 32, 34, 38 and 40 L b The factorization of 2 L The factorization of Remark 4.3. For every L {32, 34, 38, 40}, the rime we find in Table 5 may be larger than P(L). From Table 5, we have the following Lemma 4.3. Lemma 4.3. For every L {32, 34, 38, 40}, there is a generalized elite rime with elite eriod length L. Theorem 2 follows from Lemma 1.1, Lemma 4.2 and Lemma 4.3. We have known that for each even L 40, there is a generalized elite rime with elite eriod length L such that q L ( 1). But it is still an oen roblem whether for every even L there is a generalized elite rime with elite eriod length L such that q L ( 1). 5 Acknowledgement We thank the referee for kind and helful comments that imrove the resentation of the aer. References [1] A. Aigner, Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind. Monatsh. Math. 101 (1986), [2] A. Björn and H. Riesel, Factors of generalized Fermat numbers. Math. Com. 67 (1998), [3] A. Chaumont and T. Müller, All elite rimes u to 250 billion. J. Integer Seq. 9 (2006), Article [4] A. Chaumont, J. Leicht, T. Müller and A. Reinhart, The continuing search for large elite rimes. Int. J. Number Theory. 5 (2009), [5] R. Crandall and C. Pomerance, Prime Numbers, a Comutational Persective, 2nd ed., Sringer-Verlag,

13 [6] H. Dubner and Y. Gallot, Distribution of generalized Fermat rime numbers. Math. Com. 71 (2001), [7] H. Dubner and W. Keller, Factors of generalized Fermat numbers. Math. Com. 64 (1995), [8] M. Křížek, F. Luca and L. Somer, 17 Lectures on Fermat numbers. From Number Theory to Geometry, Sringer, [9] T. Müller, Searching for large elite rimes. Exeriment. Math (2006), [10] T.Müller and Andreas Reinhart, On generalized elite rimes. J. Integer Seq. 11 (2008), Article [11] H. Press, A. Teukolsky, T. Vetterling and P. Flannery, Numerical Recies in C++. The Art of Comuter Programming, Cambridge University Press, [12] H. Kenneth Rosen, Elementary Number Theory and its Alications, Addison-Wesley, Fourth edition, [13] N. J. A. Sloane, Online Encycloedia of Integer Sequences (OEIS). Electronically ublished at: htt:// njas/sequences/ 2000 Mathematics Subject Classification: Primary 11Y16; Secondary 11A15, 11A41, 11Y55. Keywords: generalized elite rimes, generalized Fermat numbers, b-fermat eriods. (Concerned with sequence A ) Received January ; revised version received June Aeared in Journal of Integer Sequences, June Return to Journal of Integer Sequences home age. 13